Right Censored Data and Empirical Likelihood
|Course||Probability Theory and Mathematical Statistics|
|Keywords||empirical likelihood right censored data EM algorithm general esti-mating equation standardχ~2 distribution|
Empirical likelihood (EL) is a nonparametric statistical method. Compared withthe normal approximation (NA) method, for complete data, the EL-based confidenceinterval doesn’t need to estimate the asymptotic variance. This is an advantage of ELmethod. But in literature, this advantage is no longer exist for right censored data. Thisdissertation focuses on constructing the EL ratio statistics of right censored data, whichconverges to the standardχ2-distribution.This dissertation consists of five chapters. In Chapter 2, EL method, empiricalprocess and semiparametric theory are introduced. Using i.i.d. complete data, thefinite sample performance of NA-based confidence intervals and EL-based confidenceintervals are compared in detail through simulation studies. The simulation resultssuggest that the confidence intervals constructed by NA and EL method are asymptoticequivalent. When the sample sizes are small, NA method and EL method have theirown advantages and disadvantages. Neither one is always better than the other.In Chapter 3, the mean residual life (MRL) inference with right censored datais considered. The likelihood ratio statistics of the MRL is constructed directly fromright censored data, and it is proved that the limiting distribution of this loglikelihoodratio statistics is standardχ2-distribution. Since this likelihood ratio statistics is hardto calculate directly, we use EM algorithm to solve the problem. Compared with theprevious results, using this loglikelihood ratio statistics to construct the confidence in-tervals doesn’t need to estimate the asymptotic variance. Simulation studies in thischapter show that the confidence intervals proposed in this chapter perform better thanthe EL-based confidence intervals and the NA-based confidence intervals.In Chapter 4, the EL inference for one-dimensional parameter based on an estimat-ing equation is discussed. The first result in this chapter is that, if the estimating equa-tion with the estimated nuisance parameter satisfies some conditions, the log EL ratiostatistics defined by this estimating equation has the limiting standardχ2-distribution.Secondly, under some assumptions, the locally most powerful log EL ratio test is thelog EL ratio statistics defined by the efficient in?uence function. At last, these theoriesare applied to right censored data. It is proved that, for right censored data, the efficientin?uence function can be used as the estimating equation to construct the log EL ratiostatistics, and the limiting distribution of this statistics is the standardχ2-distribution.Compared with the log EL ratio statistics which converges to the scaledχ2-distribution,the simulation studies show that the coverage probabilities of the confidence intervals constructed by the new method are better, and the average time spent on calculation isshorter.In Chapter 5, based on right censored data, the EL method with general estimatingequations is developed. Suppose the parameter of interest is p-dimensional, and the es-timating equations are r-dimensional, where r≥p. Qin, Lawless (1994) discussed thismodel for i.i.d. complete data. They proposed an estimator called maximum empiricallikelihood estimate for the parameter of interest, and constructed the corresponding ELratio statistics for parameter. One contribution of this chapter is to extend their theoryto right censored data. In order to define the EL ratio statistics, the estimating equationof complete data should be modified for right censored data. Inspired by the exam-ple discussed in Chapter 4, two different estimating equations are proposed, and theEL ratio statistics are defined. It is proved that the limiting distribution of these twostatistics are the scaledχ2-distribution and standardχ2-distribution, respectively. Sincethere’s no need to estimate the scaled parameter, it is more convenient to use the log ELratio statistics which converges to standardχ2-distribution to construct the EL-basedconfidence intervals. The simulation results in this chapter support the theory. And theconfidence intervals constructed by the statistics with limiting standardχ2-distributionhave better coverage accuracy than that of the other.